SOLUTION: Find the value of a if 7^a -7^a-5 = 117642√7

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Question 1204792: Find the value of a if 7^a -7^a-5 = 117642√7
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

I'll assume the equation is

It can be rewritten to and then to where

Let's solve for x.


















Notice how 7^6 = 117649
We can rewrite the right hand side as a power of 7.









plug in x = 7^a

The bases are both 7, so the exponents must be the same.
Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
.
Find the value of a if 7^a -7^a-5 = 117642√7
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Your starting equation is this

     -  = .


Factor left side by taking the factor   out parentheses.

You will get an EQUIVALENT equation

      = .


It is the same as 

     = .


Divide both sides by integer number 16806.

Notice that 117642 : 16806 = 7  with no remainder.

After dividing (canceling common factor 16806), you have

     = .


From it, you conclude that  a-5 = 1.5  (it is seen even by unarmed eye).

So, the ANSWER is : a = 5 + 1.5 = 6.5.

Solved.

This solution is short and straightforward.



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